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Wreath products and representations of $$p$$-local finite groups. (English) Zbl 1248.20019
From the introduction: Given two finite $$p$$-local finite groups and a fusion preserving morphism between their Sylow subgroups, we study the question of extending it to a continuous map between their classifying spaces. The results depend on the construction of the wreath product of $$p$$-local finite groups which is also used to study $$p$$-local permutation representations.
A fusion system $$\mathcal F$$ on a finite $$p$$-group $$P$$ is a small category whose objects are the subgroups of $$P$$ and whose morphisms are group monomorphisms which include all those homomorphisms obtained from conjugation by the elements of $$P$$. The idea of saturated fusion systems was formulated in the early 1980’s by L. Puig [J. Algebra 303, No. 1, 309-357 (2006; Zbl 1110.20011)] who studied representations of finite groups. Every block $$b$$ of the group algebra $$kG$$, where $$k$$ is an algebraically closed field of characteristic $$p$$, gives rise to a saturated fusion system on its defect group $$P\leq G$$. The principal block of $$kG$$ gives rise to the fusion system $$\mathcal F_S(G)$$ whose objects are the subgroups of a Sylow $$p$$-subgroup $$S$$ of $$G$$ and its morphisms are induced by conjugation in $$G$$.
In order to understand self-homotopy equivalences of $$BG^{\land}_p$$, C. Broto, R. Levi and B. Oliver considered [in [5] = Invent. Math. 151, No. 3, 611-664 (2003; Zbl 1042.55008)] a category $$\mathcal L_S(G)$$ closely related to $$\mathcal F_S(G)$$. This category was earlier studied by Puig. Abstraction of this construction led them [in J. Am. Math. Soc. 16, No. 4, 779-856 (2003; Zbl 1033.55010)] to the notion of a centric linking system $$\mathcal L$$ associated to a saturated fusion system $$(S,\mathcal F)$$. The triple $$(S,\mathcal F,\mathcal L)$$ is called a $$p$$-local finite group. Its classifying space is by definition the space $$|\mathcal L|^{\land}_p$$, a terminology justified by the fact that $$|\mathcal L_S(G)|^{\land}_p\simeq BG^{\land}_p$$ [5, loc. cit., Lemma 1.2]. The spaces $$|\mathcal L|^{\land}_p$$ have many properties in common with $$p$$-completed classifying spaces of finite groups. Thus, $$p$$-local finite groups provide an important connection between group theory and topology via their linking systems.
This paper focuses on the following fundamental problem. In what way, if any, a fusion preserving map $$(S,\mathcal F)\to(S',\mathcal F')$$, see details below, gives rise to a map $$|\mathcal L|^{\land}_p\to|\mathcal L'|^{\land}_p$$ between the classifying spaces? A step forward is given in Theorem B below. It is related to the yet open problem of defining the concept of morphisms between $$p$$-local finite groups in a way which is compatible with maps between their classifying spaces. It also gives a new insight to the study of maps between $$p$$-completed classifying spaces.
We will define a permutation representations of a $$p$$-local finite group $$(S,\mathcal F,\mathcal L)$$ as a homotopy class of a map $$|\mathcal L|\to(B\Sigma_n)^{\land}_p$$ where $$\Sigma_n$$ is a symmetric group. In Theorem C below we will prove a $$p$$-local form of Cayley’s theorem, namely the existence of $$p$$-regular representations. We will then approach the notion of the homotopy-index of the Sylow subgroup $$S$$ in $$(S,\mathcal F,\mathcal L)$$ through the regular representation. The index of a subgroup $$S$$ in a finite group $$G$$ is the number of orbits of $$S$$ in its action by translation of $$G$$. In other words, restriction of the regular representation of $$G$$ to $$S$$ results in $$|G:S|$$ copies of the regular representation of $$S$$. From the homotopy point of view, one could define the homotopy-index of $$S$$ in $$\mathcal L$$ by the minimal $$n$$ for which there is a map $$|\mathcal L|\to(B\Sigma_{n\cdot|S|})^{\land}_p$$ whose restriction to $$BS$$ is homotopic to the map $$BS@>n\cdot\text{reg}_S>>(B\Sigma_{n\cdot|S|})^{\land}_p$$ induced by $$n$$ copies of the regular representation of $$S$$. But this number is very difficult to compute, even for a $$p$$-local finite group associated to a finite group. Instead, we will define the lower homotopy-index of $$S$$ in $$\mathcal L$$ as the smallest number $$p^k$$ such that the map $$BS\to(B\Sigma_{p^k\cdot|S|})^{\land}_p$$ induced by $$p^k\cdot\text{reg}_S$$ can be extended up to homotopy to a map $$|\mathcal L|\to(B\Sigma_{p^k\cdot|S|})^{\land}_p$$. This is a new invariant of $$p$$-local finite groups.

##### MSC:
 20D20 Sylow subgroups, Sylow properties, $$\pi$$-groups, $$\pi$$-structure 55R35 Classifying spaces of groups and $$H$$-spaces in algebraic topology 55R37 Maps between classifying spaces in algebraic topology 55R40 Homology of classifying spaces and characteristic classes in algebraic topology 20D30 Series and lattices of subgroups 20E22 Extensions, wreath products, and other compositions of groups
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